We present the full source catalogue from the Australia Telescope 20 GHz (AT20G) Survey. The AT20G is a blind radio survey carried out at 20 GHz with the Australia Telescope Compact Array (ATCA) from 2004 to 2008, and covers the whole sky south of declination 0 deg. The AT20G source catalogue presented here is an order of magnitude larger than any previous catalogue of high-frequency radio sources, and includes 5890 sources above a 20 GHz flux-density limit of 40 mJy. All AT20G sources have total intensity and polarisation measured at 20 GHz, and most sources south of declination -15 deg also have near-simultaneous flux-density measurements at 5 and 8 GHz. A total of 1559 sources were detected in polarised total intensity at one or more of the three frequencies. We detect a small but significant population of non-thermal sources that are either undetected or have only weak detections in low-frequency catalogues. We introduce the term Ultra-Inverted Spectrum (UIS) to describe these radio sources, which have a spectral index alpha(5, 20) > +0.7 and which constitute roughly 1.2 per cent of the AT20G sample. The 20 GHz flux densities measured for the strongest AT20G sources are in excellent agreement with the WMAP 5-year source catalogue of Wright et al. (2009), and we find that the WMAP source catalogue is close to complete for sources stronger than 1.5 Jy at 23 GHz.
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